The Landau-Siegel zero problem: a uniform logarithmic gap for real zeros of real Dirichlet L-functions

For a primitive Dirichlet character $\chi$ of conductor $q$ , the classical zero-free region $\Re s\ge1-c_1/\log(q(|\Im s|+2))$ holds with one possible exception: a single simple real zero $\beta$ attached to a real character, which could lie very close to $1$ . Siegel's theorem (1935) gives $L(1,\chi)\gg_\varepsilon q^{-\varepsilon}$ only ineffectively, and Page's theorem limits such zeros to at most one per range of conductors, so a sequence of real zeros with $(1-\beta)\log q\to0$ is not ruled out. In its logarithmic formulation the Landau-Siegel zero problem asks: is there an absolute constant $c>0$ such that every real zero $\beta\in(0,1)$ of every primitive nonprincipal real Dirichlet $L$ -function of conductor $q\ge3$ satisfies $(1-\beta)\log q\ge c$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Analytic number theory
Posed by: Classical exceptional-zero problem going back to Landau, Page and Siegel; logarithmic zero-gap conjecture as discussed by Friedlander and Iwaniec (2018)
Year posed: —
Years open: —
Solved: 2026-09-30
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 72 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Claims an absolute constant $c>0$ such that every real zero $\beta\in(0,1)$ of every primitive nonprincipal real Dirichlet $L$ -function of conductor $q\ge3$ satisfies $(1-\beta)\log q\ge c$ . The proof combines a prime-bias estimate forced by a zero close to $1$ with an interpolation determinant on a biquadratic field $\mathbb Q(\sqrt d,\sqrt2)$ . The constant is not made explicit, so the result does not by itself give effective class-number bounds with numbers attached. It excludes real zeros in $1-c/\log q<\beta<1$ only; zeros elsewhere in $(0,1)$ are not addressed by this manuscript (the family's 7/8 manuscript excludes those above $7/8$ ).

What the AI did

The release README says every manuscript in the collection was produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result. It lists work on a zero-free region for the Riemann zeta function as an exception to that procedure; whether this companion on Landau-Siegel zeros falls under the exception is not stated. The manuscript is credited to OpenAI with no human author named. The same conclusion also follows from the family's 7/8 half-plane, which removes every real zero above 7/8.

Verification

No independent mathematician has checked this yet. Theorem 1 was read against the posed problem: it is the logarithmic formulation exactly, with an absolute but inexplicit constant. formalization.yaml lists OAI.SiegelZeros.WeightedTorusJets.exists_absolute_real_zero_gap as a main result for this manuscript. Its comparator statement uses Mathlib's DirichletCharacter.LFunction and asserts a c > 0 such that for every q >= 3 and every primitive nonprincipal character with real values and every real zero 0 < beta < 1, c <= (1 - beta) log q. That is the headline claim; both parities are included and no value of c is given. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The proof is short (an interpolation-determinant argument in a biquadratic field) and independent of the quasi-Riemann manuscript, though that manuscript implies the same statement.

The Landau-Siegel zero problem: a uniform logarithmic gap for real zeros of real Dirichlet L-functions

What was actually shown

What the AI did

Verification

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Discussion